Theorem eldmcoss 39217

Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)

Assertion

Ref	Expression
eldmcoss	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

Distinct variable groups:   𝑢,𝐴   𝑢,𝑅   𝑢,𝑉

Proof of Theorem eldmcoss

Step	Hyp	Ref	Expression
1		dmcoss3 39212	. . 3 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2	1	eleq2i 2855	. 2 ⊢ (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ∈ dom ◡𝑅)
3		eldmcnv 39014	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
4	2, 3	bitrid 286	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 209  ∃wex 1809   ∈ wcel 2143   class class class wbr 5109  ^◡ccnv 5660  dom cdm 5661   ≀ ccoss 38852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-coss 39170

This theorem is referenced by:  eldmcoss2  39218  eldm1cossres  39219